On the other end of the dynamical spectrum lie hyperbolic actions, especially actions with an Anosov element. By this one usually means that some element \(g \in G\)  acts normally hyperbolically w.r.t. the orbit foliation of an associated subgroup of \(G\)  (eg, the centralizer of an \(\mathbb {R}\) -split Cartan subgroup in a semisimple Lie group). That is, \(g\)  uniformly expands and contracts complementary subbundles transverse to the orbit foliation of the associated subgroup. For lattices \(\Gamma \)  this is simply that some element \(\gamma \in \Gamma \)  acts via an Anosov diffeomorphism.
Examples of such actions are known to exist on certain nilmanifolds \(N/\Lambda \)  and finite quotients of such where \(\Lambda \)  is a lattice in a (simply connected) nilpotent group \(N\) . Conversely, if the underlying manifold is a nilmanifold,
Brown, Rodriguez Hertz and Wang prove the beautiful global rigidity result that all actions of higher rank cocompact lattices are \(C^{\infty }\) -conjugate to one by automorphisms [1]}. For non-uniform \(\Gamma \) , they need to assume additional conditions such as being able to lift the action to the universal cover of \(N/\Lambda \) . In addition, a well-known 50 year old question of Anosov and Smale asks whether Anosov diffeomorphisms only exist on nilmanifolds and their finite quotients [2]}. A positive answer to this question would immediately imply the classification of Anosov actions at least for irreducible higher rank uniform lattices. However, at this point this question remains wide open.
